Typical Cases

Visualisations

Temperature fields at the end of the simulations.

2D:

3D:

Incompressibility constraint

Maximum divergence of the velocity field, which should be sufficiently small.

2D:

3D:

Nusselt numbers

Evolution

\(Nu\) calculated using the different formulae, which are monitored during the run, are shown as a function of time:

• red: heat fluxes on the walls

• blue: energy input

• green: kinetic energy dissipation

• magenta: thermal energy dissipation

2D:

3D:

Note

The black-dashed line in the two-dimensional result shows a reference value by van der Poel et al., J. Fluid Mech. (736), 2013 with the same \(Ra\) and \(Pr\) but the different domain geometry is different (box).

See also

Nusselt number relations.

Temporary-Averaged Values

As derived here, there are two contributions which transfer heat: advective contribution:

\[\sumzc \sumyc \frac{J}{\sfact{1}} \vel{1} \ave{T}{\gcs{1}},\]

and diffusive contribution:

\[- \sumzc \sumyc \frac{1}{\sqrt{Pr} \sqrt{Ra}} \frac{J}{\sfact{1}} \frac{1}{\sfact{1}} \dif{T}{\gcs{1}}.\]

After averaged over time and homogeneous directions, they are displayed as a function of the wall-normal position \(x\) here:

2D:

3D:

Standard deviations

Variances of (red) \(\ux\), (blue) \(\uy\), (magenta) \(\uz\), and (green) \(T\) are shown here.

2D:

3D:

Note

Although the \(y\) and the \(z\) directions are homogeneous, the blue and magenta lines may deviate, which is attributed to the low \(Ra\) and short time.